Inference for sparse and dense functional data with covariate adjustments

Abstract

We consider inference for the mean and covariance functions of covariate adjusted functional data using Local Linear Kernel (LLK) estimators. By means of a double asymptotic, we differentiate between sparse and dense covariate adjusted functional data -depending on the relative order of m (the discretization points per function) and n (the number of functions). Our simulation results demonstrate that the existing asymptotic normality results can lead to severely misleading inferences in finite samples. We explain this phenomenon based on our theoretical results and propose finite-sample corrections which provide practically useful approximations for inference in sparse and dense data scenarios. The relevance of our theoretical results is showcased using a real-data application.

Low-Rank Covariance Function Estimation for Multidimensional Functional Data

“…Remark 6. For covariance function estimation for unidimensional functional data, i.e., p = 1, a limited number of approaches, including Cai and Yuan (2010), Li and Hsing (2010), Wang (2016), andLiebl (2019), can achieve unified theoretical results in the sense that they hold for all relative magnitudes of n and m. The similarity of these approaches is the availability of a closed form for each covariance function estimator. In contrast, our estimator obtained from (7) does not have a closed form due to the non-differentiability of the penalty, but it can still achieve unified theoretical results which hold for both unidimensional and multidimensional functional data.…”

“…It automatically incorporates the settings of dense and sparse functional data, and reveals a phase transition in the rate of convergence. Different from existing theoretical work heavily based on closed-form representations of estimators, (Li and Hsing, 2010;Cai and Yuan, 2010;Zhang and Wang, 2016;Liebl, 2019), this paper provides the first unified theory for penalized global M-estimators of covariance functions which does not require a closed-form solution. Furthermore, a near-optimal (i.e., optimal up to a logarithmic order) one-dimensional nonparametric rate of convergence is attainable for the 2p-dimensional covariance function estimator for Sobolev-Hilbert spaces.…”

Nonparametric Testing for Differences in Electricity Prices: The Case of the Fukushima Nuclear Accident

“…The smoothness of the underlying price curve X i induces a high correlation between electricity prices Y ij and Y ik with similar values of electricity demand U ij ≈ U ik . Ignoring these correlations when doing inference can result in serious size distortions and invalid test decisions (see Liebl, 2019a).…”

“…which leads to the above mentioned functional-data-specific variance term that makes it necessary to use the finite sample correction proposed by (Liebl, 2019a).…”

“…The following theorem is an adjusted version of Corollary 3.1, part (b) in Liebl (2019a) and takes into account the finite sample correction for the variance component:…”